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Mirrors > Home > MPE Home > Th. List > rankdmr1 | Structured version Visualization version GIF version |
Description: A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankdmr1 | β’ (rankβπ΄) β dom π 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankidb 9799 | . . . 4 β’ (π΄ β βͺ (π 1 β On) β π΄ β (π 1βsuc (rankβπ΄))) | |
2 | elfvdm 6928 | . . . 4 β’ (π΄ β (π 1βsuc (rankβπ΄)) β suc (rankβπ΄) β dom π 1) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π΄ β βͺ (π 1 β On) β suc (rankβπ΄) β dom π 1) |
4 | r1funlim 9765 | . . . . 5 β’ (Fun π 1 β§ Lim dom π 1) | |
5 | 4 | simpri 485 | . . . 4 β’ Lim dom π 1 |
6 | limsuc 7842 | . . . 4 β’ (Lim dom π 1 β ((rankβπ΄) β dom π 1 β suc (rankβπ΄) β dom π 1)) | |
7 | 5, 6 | ax-mp 5 | . . 3 β’ ((rankβπ΄) β dom π 1 β suc (rankβπ΄) β dom π 1) |
8 | 3, 7 | sylibr 233 | . 2 β’ (π΄ β βͺ (π 1 β On) β (rankβπ΄) β dom π 1) |
9 | rankvaln 9798 | . . 3 β’ (Β¬ π΄ β βͺ (π 1 β On) β (rankβπ΄) = β ) | |
10 | limomss 7864 | . . . . 5 β’ (Lim dom π 1 β Ο β dom π 1) | |
11 | 5, 10 | ax-mp 5 | . . . 4 β’ Ο β dom π 1 |
12 | peano1 7883 | . . . 4 β’ β β Ο | |
13 | 11, 12 | sselii 3979 | . . 3 β’ β β dom π 1 |
14 | 9, 13 | eqeltrdi 2840 | . 2 β’ (Β¬ π΄ β βͺ (π 1 β On) β (rankβπ΄) β dom π 1) |
15 | 8, 14 | pm2.61i 182 | 1 β’ (rankβπ΄) β dom π 1 |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wb 205 β wcel 2105 β wss 3948 β c0 4322 βͺ cuni 4908 dom cdm 5676 β cima 5679 Oncon0 6364 Lim wlim 6365 suc csuc 6366 Fun wfun 6537 βcfv 6543 Οcom 7859 π 1cr1 9761 rankcrnk 9762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-r1 9763 df-rank 9764 |
This theorem is referenced by: r1rankidb 9803 pwwf 9806 unwf 9809 uniwf 9818 rankr1c 9820 rankelb 9823 rankval3b 9825 rankonid 9828 rankssb 9847 rankr1id 9861 |
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